To find the area of triangle ATUV given two sides and the included angle, we can use the formula for the area of a triangle when you know two sides and the included angle between them. This formula is:
\(\[ \text{Area} = \frac{1}{2} uv \sin(M) \]\)
where:
- \(\( u \)\) is one side of the triangle,
- \(\( v \)\) is the other side,
- \(\( M \)\) is the angle between sides \(\( u \)\) and \(\( v \)\).
Here are the given values:
- \(\( M = 68^\circ \)\)
- \(\( u = 5 \)\) cm
- \(\( v = 6 \)\) cm
Step-by-step solution:
1. Convert the angle \(\( M \)\) from degrees to radians. Since trigonometric functions in mathematical formulas generally use radians, we must convert degrees to radians by using the conversion factor:
\(\[ 1^\circ = \frac{\pi}{180} \]\)
So:
\(\[ M \text{ (in radians)} = 68^\circ \times \frac{\pi}{180} \approx 1.1868 \text{ radians} \]\)
2. Use the sine of the angle \(\( M \)\). We need to find \(\( \sin(68^\circ) \)\). If you use a calculator, it will give you approximately:
\(\[ \sin(68^\circ) \approx 0.9272 \]\)
3. Plug the values into the area formula:
\(\[ \text{Area} = \frac{1}{2} \times 5 \text{ cm} \times 6 \text{ cm} \times \sin(68^\circ) \]\)
\(\[ \text{Area} = \frac{1}{2} \times 5 \times 6 \times 0.9272 \]\)
4. Compute the product:
\(\[ \text{Area} = \frac{1}{2} \times 5 \times 6 \times 0.9272 \approx \frac{1}{2} \times 5 \times 6 \times 0.9272 = 13.908 \]\)
5. Round the result to the nearest tenth:
\(\[ \text{Area} \approx 13.9 \, \text{square centimeters} \]\)
Thus, the area of triangle ATUV is approximately \(\( 13.9 \)\) square centimeters.
